This project seeks to estimate sport fish harvest and releases of rockfish in Alaska waters by improving on the Howard et al. (2020) methods and expand the time series back to 1977 when the statewide harvest survey (SWHS) was first implemented. This is essentially a Bayesian version of the Howard methods that allows for more appropriate and defensible sharing of information between areas, handles missing data in a more appropriate manor, accurately propagates uncertainty throughout the estimation procedure and replaces the Howard decision tree approach to low sample sizes with a hierarchical model. The methods and results for generating harvest estimates are generally consistent between the Bayesian model and the Howard methods. Harvest estimates are consistent with Howard estimates during contemporary times, but may differ based on more appropriate weighting of SWHS and logbook data, including estimating and correcting bias in the SWHS data.

The Bayesian methods depart from the Howard method in how releases are estimated. The Howard methods assume that the species composition of the harvests are equal to the species composition of released fish, which is clearly contraindicated in the logbook data. For instance, logbook data demonstrates that yelloweye have been retained at high levels up until restrictions were enacted in recent years, whereas pelagic rockfish were released in significant numbers in the past with retention increasing in recent years as they have become more prized by anglers. Recent prohibition on retaining yelloweye in Southeast Alaska highlights the shortcomings of the original Howard assumptions as the species composition of the harvest would indicate that no yelloweye were caught and released during the closure.

The Howard method for estimating releases for private anglers also relied on an expansion of the logbook release estimates based on the ratio of private:guided releases of all rockfish in the SWHS. In addition to the faulty assumptions about species composition, this method ignores potential bias in SWHS estimates of harvests and releases or at least assumes that the bias in release and harvests are the same. As demonstrated in Figure 1, the bias in those two quantities appears to be quite different based on the logbook data. The Bayesian model thus attempts to estimate release probabilities based on the logbook data coupled with bias corrected estimates from the SWHS.

Lastly, the Howard methods were only used on data beginning in 1999 with the advent of the logbook program and estimates of harvests and releases prior to that have been based on linear ramps from 1999 back to the perceived start of the fishery. The Bayesian methods allow us to expand the time series back to 1977 when the SWHS was implemented by leveraging regional data trends in species composition and the proportion of caught rockfish harvested by species and/or species complex. Key advantages of the Bayesian approach are highlighted in table 1.

Table 1. Summary of key improvements in reconstructiing sport fish removals of rockfish using the Bayesian model as compared to the Howard et al. (2020) methods.
Issue Howard Bayes
Time series 1999 - present 1977 - present
Bias in SWHS Not explicitly dealt with. Relies on logbook data and ratios of guided/unguided from SWHS data to estimate unguided releases and harvests. Explicitly estimates bias in SWHS harvest and release estimates based on logbook data.
Species composition of releases Assumes that species composition of releases is equal to that of the harvest, which is not evident in the logbook data. Recognizes different release probabilities by species / species assemblage and estimates it from logbook data and bias corrected SWHS data
Sample size limitations Uses sample size threshholds such that when areas fall below those threshholds values are borrowed from nearby areas. Uses a hierarchichacal modelling approach that shares information between areas in the same region. Thus all data is used, even with small sample sizes. This is a more sound method that avoids assumptions and uses all of the data.
Error propogation Error is propogated when variance estimates are available, but there is uncertainty associated with borrowing values from nearby areas, or the assumption of species compositions being identical in harvest and releases, are not dealt with. By breaking the assumption that species composition is equal between harvests and releases, uncertainty in the release estimates is more reflective of the fishery. Furthermore, the hyerarchichal approach more accurately captures uncertainy within and between areas within a region.

Data

Harvest data was available for 22 commercial fishing management areas in Southcentral and Southeast Alaska. Areas with negligible rockfish harvest were pooled with adjacent areas for analysis. Specifically the Aleutian and Bering areas were pooled into an area labeled BSAI; the IBS and EKYT were pooled into an area labeled EKYKT; the Southeast, Southwest, SAKPEN and Chignik areas were pooled into an area labeled SOKO2PEN and the Westside and Mainland areas were pooled into an area labeled WKMA.

Stateside Harvest Survey (SWHS)

Statewide harvest survey estimates of rockfish catch and harvest are available for 28 years (1996-2023) for all users and for 13 years (2011-2023) for guided anglers (Figure 0). Additionally, there are overall harvest estimates from 1977- 1995 and release estimates from 1990-1995 that required some partitioning to ascribe to current management units. Harvests in unknown areas were apportioned based on harvest proportions in 1996. Variance estimates are not available for pre-1996 data and as such, the maximum observed coefficient of variation (cv) in each commercial fisheries management unit was applied to the pre-1996 values.

**Figure 1.**- Data sources for estimating rockfish harvests and releases in ADF&G commercial fisheries management units. Note that initial rockfish harvest estimates were not differentiated into species assemblage or species until 1998 when logbooks began differentiating by pelagic and non-pelagic. Logbooks began to collect data on yelloweye beginning in 2006. Port sampling programs to gather data on species composition of harvests began in 1996 in Southcentral and Kodiak and in 2006 in Southeast.

Figure 1.- Data sources for estimating rockfish harvests and releases in ADF&G commercial fisheries management units. Note that initial rockfish harvest estimates were not differentiated into species assemblage or species until 1998 when logbooks began differentiating by pelagic and non-pelagic. Logbooks began to collect data on yelloweye beginning in 2006. Port sampling programs to gather data on species composition of harvests began in 1996 in Southcentral and Kodiak and in 2006 in Southeast.


SWHS estimates are believed to be biased to some degree. These modelling efforts aim to estimate and correct for that bias with the assumption that logbook records are a census of guided harvests and releases.

SWHS Rockfish release estimates are inferred from the difference between catch and harvest estimates.

Adam noted that the first 5 years (23 years counting the historical data) in the SWHS data set for PWSO seem unreasonable (close to zero and not corroborated with logbook estimates). Adam recommended setting these harvests to unknown, but current model development has included the data. Once a satisfactory model has been identified we will exam the effects of censoring the PWSO data.

Creel Surveys

NA

Guide Logbooks

Sport fishing guides have been required to report their harvest of rockfish for 26 years (1998-2023). Reported harvest is also available by assemblage (pelagic vs. non-pelagic). Harvest of yelloweye and “other” (non-pelagic, non-yelloweye) rockfish were reported separately beginning in 2006.

Logbooks also record the number of rockfish released for the same categories. However, the reliability of the release data is somewhat questionable as reported releases are generally far lower than that estimated by the SWHS. As such several treatments of the data are considered.

Logbook versus SWHS estimates

Estimates of guided harvests and releases from the SWHS do not align with the census from charter logbooks. Logbook harvest reports are generally considered reliable and are used to assess the bias in SWHS reports. However, there is even greater disparity between release estimates in the two sources and it is debatable whether logbook releases should be treated as a census. The Howard et al. (2020) methods do treat the logbook release data as “true” and thus are considerably less than would be estimated from the SWHS data.

**Figure 2.**- SWHS harvest (left) and release (right) estimates from guided trips (x-axis) versus repoted harvests from charter logbooks (y-axis).

Figure 2.- SWHS harvest (left) and release (right) estimates from guided trips (x-axis) versus repoted harvests from charter logbooks (y-axis).


A note on model development

To evaluate the discrepancy in apparent bias in harvest and release data, several models were explored to estimate releases during model development. One method (\(LB_{fit}\)) considers the logbook release data to be reliable and a second method (\(LB_{cens}\)) treated the logbook release data as estimates of the minimum released, thus giving more weight to SWHS release estimates. A third method (\(LB_{hyb}\)) is a hybrid approach that treats reported releases of yelloweye as reliable but total rockfish and pelagic rockfish releases as minimums. Model development revealed a tension between the total and pelagic logbook releases and the yelloweye logbook releases. This tensions eventually highlighted the different release/retention probabilities between yelloweye and pelagics in the logbook data and prompted the current approach whereby that probability was calculated for the three main species complexes covered in the data: pelagics, yelloweye, and “other”. The methods described here follow the (\(LB_{fit}\)) formulation. Based on model behavior it is unlikely that the (\(LB_{cens}\)) model would work as there would not be enough data to estimate release probabilities. However, it may be worth running the (\(LB_{hyb}\)) approach as a sensitivity test at the very least.

Composition data

Harvest sampling data exists from Gulf of Alaska areas since 1996 and from Southeast Alaska areas since 2006. Port sampling data is comprised of the number of total rockfish, pelagic and non-pelagic rockfish, black rockfish and yelloweye rockfish. In Southeast Alaska, the number of Demersal Shelf Rockfish (DSR, of which yelloweye are one species) and slope rockfish are also recorded.

Process equations

The true harvest \(H_{ay}\) of rockfish for area \(a\) during year \(y\) is assumed to follow a temporal trend defined by a penalized spline:

\[\begin{equation} \textrm{log}(H_{ay})~\sim~\textrm{Normal}(f(a,y), {\sigma_H}) \end{equation}\]

where \(f(a,y)\) in a p-spline basis with 7 components (knots) and a second degree penalty. The variance, \(\sigma_H\), was given a normal prior with a mean and standard deviation of 0.25 and 1, respectively.

Charter and private harvest \(H_{ayu}\) (where u = 1 for charter anglers and u = 2 for private anglers) is a fraction of total annual harvest in each area:

\[\begin{equation} H_{ay1}~=~H_{ay}P_{(user)ay1}\\H_{ay2}~=~H_{ay}(1-P_{(user)ay1}) \end{equation}\]

where \(P_{(user)ay1}\) is the fraction of the annual harvest in each area taken by charter anglers. \(P_{(user)ay1}\) was modeled hierarchically across years as:

\[\begin{equation} P_{(user)ay1}~\sim~\textrm{beta}(\lambda1_a, \lambda2_a) \end{equation}\]

with non-informative priors on both parameters.

Annual black rockfish harvest \(H_{(black)ayu}\) for each area and user group is:

\[\begin{equation} H_{(black)ayu}~=~H_{ayu}P_{(pelagic)ayu}P_{(black|pelagic)ayu} \end{equation}\]

where \(P_{(pelagic)ayu}\) is the fraction of the annual harvest for each area and user group that was pelagic rockfish and \(P_{(black|pelagic)ayu}\) is the fraction of the annual harvest of pelagic rockfish for each area and user group that was black rockfish.

The southeast region also tracks two other non-pelagic rockfish assemblages, demersal shelf rockfish (DSR, which includes yelloweye) and slope rockfish. For the southeast region the harvest of those two assemblages is thus

\[\begin{equation} H_{(DSR)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(DSR|non-pelagic)ayu}\\ H_{(slope)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(slope|non-pelagic)ayu}\\ \end{equation}\]

where \(P_{(DSR|non-pelagic)ayu}\) and \(P_{(slope|non-pelagic)ayu}\) are the fractions of the annual harvest of non-pelagic rockfish for each area and user group that were DSR and slope rockfish, respectively.

Annual yelloweye rockfish harvest \(H_{(yelloweye)ayu}\) for each area and user group is calculated differently for central/Kodiak areas and southeast areas. For central and Kodiak areas yelloweye rockfish harvests are calculated as

\[\begin{equation} H_{(yelloweye)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(yelloweye|non-pelagic)ayu} \end{equation}\]

where \(P_{(yellow|non-pelagic)ayu}\) is the fraction of the annual harvest of non-pelagic rockfish for each area and user group that was yelloweye rockfish.

For southeast areas yelloweye harvests are a fraction of the DSR harvests such that

\[\begin{equation} H_{(yelloweye)ayu}~=~H_{(DSR)ayu}P_{(yelloweye|DSR)ayu} \end{equation}\]

The composition parameters \(P_{(comp)ayu}\), were modeled using a logistic curve that would allow hindcasting without extrapolating beyond the limit of observed values such that:

\[\begin{equation} \textrm{logit}(P_{(comp)ayu})~=~\beta0_{(comp)ayu} + \frac{\beta1_{(comp)ayu}}{(1 + exp(\beta2_{(comp)ayu}*(y - \beta3_{(comp)ayu})))} + \beta4_{(comp)ayu}*I(u=private)+re_{(comp)ayu} \end{equation}\]

where the \(\beta\) parameters define the intercept, scaling factor, slope, inflection point and private angler effect, respectively, \(y\) is the year index, \(I(u=private)\) is an index variable which is 1 when the user groups is private and 0 otherwise and \(re_{(comp)ayu}\) is a random effect with a non-informative prior. \(\beta\) parameters were modeled hierarchically by region. When \(\beta\) parameters were inestimable as a result of no discernible change in composition over the observed time period. \(\beta1\) (scaling factor) and \(\beta2\) (slope) were fixed to 0 so that the long term mean value was used for hindcasting.

The true number of released rockfish \(R_{ayu}\) were based on the proportion of the total catch harvested, \(pH_{(comp)ayu}\), by area, year, user group and species grouping. Because release data from the SWHS is for all rockfish and the release data from logbooks is only subdivided into pelagics, yelloweye and “other” (non-pelagic, non-yelloweye), we only estimated \(pH_{(comp)ayu}\) for those categories. Thus, converting \(H_{(comp)ayu}\) to total catches by user group, \(C_{(comp)ayu}\), with \(pH_{(comp)ayu}\) results in estimates of total releases such that

\[\begin{equation} R_{(comp)ayu}~=~ C_{(comp)ayu} - H_{(comp)ayu} ~=~ \frac{H_{(comp)ayu}}{pH_{(comp)ayu}} - H_{(comp)ayu} \end{equation}\]

with total releases equal to the sum of the compositional releases. For non-yelloweye DSR and Slope rockfish assemblages in Southeast Alaska \(R_{(DSR)ayu}\) and \(R_{(slope)ayu}\) were estimated from \(R_{(other)ayu}\) using the species composition data from the harvest, thus assuming that slope and DSR assemblages were caught and released at the same rates.

The proportion harvest parameters for \(pH_{(comp)ayu}\) were modeled using a logistic curve that would allow hindcasting based on trends in the data without extrapolating beyond the range of observed values such that

\[\begin{equation} \textrm{logit}(pH_{(pH)ayuc})~=~\beta0_{(pH)ayu} + \frac{\beta1_{(pH)ayuc}}{(1 + exp(\beta2_{(pH)ayuc}*(y - \beta3_{(pH)ayuc})))} + \beta4_{(pH)ayuc}*I(u=private)+re_{(pH)ayuc} \end{equation}\]

A random effect term allowed estimation during the historical period when data is available, but the curve defined by the above equation determined release estimates between 1977 and 1990. As with the compositional trends, \(\beta\) parameters were modeled hierarchically by region. When \(\beta\) parameters were inestimable as a result of no discernable change in harvest probability over the observed time period, \(\beta1\) (scaling factor) and \(\beta2\) (slope) were fixed to 0 so that the long term mean value was applied.

Release mortality (i.e., the number of released rockfish expected to die) was calculated assuming fixed mortality rates developed in each of the regions. Deep water release (DWR) devices were mandated for charter fleets in 2013 and rates were derived from CITATION. Southeast applies basic rates estimated in these studies while Southcentral and Kodiak rates were derived by using historical depth-of-release data to adjust the rates based on area and user group.

The total number of mortalities by year, area, user and species/species assemblage in numbers was calculated by summing harvests and release mortality such that

\[\begin{equation} M_{(comp)ayu}~=~ H_{(comp)ayu} + m_{R-(comp)ayu} * R_{(comp)ayu} \end{equation}\]

where \(m_{R-(comp)ayu}\) is the release mortality rate by year, area, user and species (Figure XX).

Total removals in biomass were converted using the average weight of fish from port sampling?. A minimum sample size per year of X fish was used as the cutoff for including in the data set. Weights were modeled hierarchically to estimate weights in years when data was missing. The total biomass of removals by year, area, user and species was thus

\[\begin{equation} B_{(comp)ayu}~=~ \overline{wt}_{(comp)ayu} * M_{(comp)ayu} \end{equation}\]

where \(\overline{wt}_{(comp)ayu}\) is the mean weight by species, area, user and year.

Observation equations

SWHS estimates of annual rockfish harvest \(\widehat{SWHS}_H{ay}\) were assumed to index true harvest:

\[\begin{equation} \widehat{SWHS}_H{ay}~\sim~\textrm{LogNormal}\left(\textrm{log}(H_{ay}b_{ay}), \sigma_{SWHSHay}^2\right) \end{equation}\]

where bias in the SWHS harvest estimates \(b_H{ay}\) is modeled hierarchically across years as:

\[\begin{equation} b_H{ay}~\sim~\textrm{Normal}(\mu_H{(b)a}, \sigma_H{(b)a}) \end{equation}\]

with non-informative priors on both parameters.

SWHS estimates of guided angler harvest \(\widehat{SWHS}_H{ay1}\) are related to total harvest by:

\[\begin{equation} \widehat{SWHS}_H{ay1}~\sim~\textrm{LogNormal}\left(\textrm{log}(H_{ay1}b_{ay}), \sigma_{SWHS_{ay1}}^2\right) \end{equation}\]

Reported guide logbook harvest \(\widehat{LB}_H{ay}\) is related to true harvest as:

\[\begin{equation} \widehat{LB}_H{ay}~\sim~\textrm{Poisson}(H_{ay1})\\ \widehat{LB}_H{(pelagic)ay}~\sim~\textrm{Poisson}(H_{ay1}P_{(pelagic)ay1})\\ \widehat{LB}_H{(yelloweye)ay}~\sim~\textrm{Poisson}(H_{(yelloweye)ay1})\\ \widehat{LB}_H{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(H_{(nonpel,nonye)ay1})\\ \end{equation}\]

Note that for central and Kodiak areas \(H_{(nonpel,nonye)ay1}\) is equal to the total harvest minus pelagic and yelloweye harvests. For southeast areas \(H_{(nonpel,nonye)ay1}\) is equal to the sum of the DSR and slope harvests minus yelloweye harvests.

SWHS estimates of annual rockfish releases \(\widehat{SWHS}_R{ay}\) were assumed to index true releases in a similar fashion and thus modeled similarly. As such, the release data are related to true releases just as harvests were modeled such that:

\[\begin{equation} \widehat{LB}_R{ay}~\sim~\textrm{Poisson}(R_{ay1})\\ \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{Poisson}(R_{ay1}P_{(pelagic)ay1})\\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

Because logbook release data is more questionable and demonstrates greater disagreement with SWHS estimates (Figure 1), a second approaches was considered that loosened the assumption that logbook releases were a census. Methods explored to develope \(LB_{hyb}\) and \(LB_{cens}\) models are detailed at the end of this section.

SWHS estimates of guided angler release \(\widehat{SWHS}_R{ay1}\) is modeled the same as harvests.

SWHS release bias was modeled independently of the harvest bias \(b_H{ay}\) such that

\[\begin{equation} b_R{ay}~\sim~\textrm{Normal}(\mu_R{(b)a}, \sigma_R{(b)a}) \end{equation}\]

where bias in the SWHS release estimates \(b_R{ay}\) is modeled hierarchically across years as:

\[\begin{equation} b_R{ay}~\sim~\textrm{Normal}(\mu_R{(b)a}, \sigma_R{(b)a}) \end{equation}\]

with non-informative priors on both parameters.

The number of pelagic rockfish sampled in harvest sampling programs \(x_{(pelagic)ayu}\) follow a binomial distribution:

\[\begin{equation} x_{(pelagic)ayu}~\sim~\textrm{Binomial}(P_{(pelagic)ayu}, N_{ayu}) \end{equation}\]

where \(N_{ayu}\) is the total number of rockfish sampled in area \(a\) during year \(y\) form user group \(u\). The number of black rockfish sampled in harvest sampling programs was thus a proportion of the pelagic harvests

\[\begin{equation} x_{(black)ayu}~\sim~\textrm{Binomial}(P_{(black)ayu}, N_{ayu}^{pel}) \end{equation}\]

Yelloweye rockfish in Southcentral and Kodiak were modeled similarly as a proportion of the total number of non-pelagics such that

\[\begin{equation} x_{(yellow_{R2})ayu}~\sim~\textrm{Binomial}(P_{(yellow_{R2})ayu}, N_{ayu}^{nonpel}) \end{equation}\]

Southeast areas have several other non-pelagic groupings such that DSR and slope rockfish are a proportion of non-pelagics

\[\begin{equation} x_{(DSR)ayu}~\sim~\textrm{Binomial}(P_{(DSR)ayu}, N_{ayu}^{nonpel}) \end{equation}\]

and

\[\begin{equation} x_{(slope)ayu}~\sim~\textrm{Binomial}(P_{(slope)ayu}, N_{ayu}^{nonpel}) \end{equation}\]

with yelloweye in southeast a proportion of the DSR harvest

\[\begin{equation} x_{(yellow_{R1})ayu}~\sim~\textrm{Binomial}(P_{(yellow_{R1})ayu}, N_{ayu}^{DSR}). \end{equation}\].

Kodiak has limited port sampling beyond the main harbors but has a robust hydroacoustic survey that is used to quantify black rockfish abundance across the management area and uses stereocameras to derive species compositions of the hydroacoustic data. This data was used as supplementary data to further inform the model to the proportion of pelagic rockfish that are black in Kodiak areas. Angler landings in Kodiak show a higher proportion of black rockfish relative to the hydroacoustic survey and thus the proportion of black rockfish in the hydroacoustic sample related to the true proportion such that

\[\begin{equation} P_{(black|pelagic)ayu}^{HA} ~\sim~ P_{(black|pelagic)ayu} + ae_{au} \end{equation}\].

where \(ae_{au}\) is the angler effect for each area and user group modeled hierarchically around a mean of 0. Predicted \(P_{(black|pelagic)ayu}^{HA}\) assumed a beta distribution such that

\[\begin{equation} P_{(black|pelagic)ayu}^{HA} ~\sim~ beta(\alpha_{HA},\beta_{HA}) \end{equation}\]

where

\[\begin{equation} \alpha_{HA} ~=~ (P_{(black|pelagic)ayu}^{HA})^2 * \frac{1 - P_{(black|pelagic)ayu}^{HA}}{\frac{var_{P_{HA}}-1}{P_{(black|pelagic)ayu}^{HA}}}, \end{equation}\]

\[\begin{equation} \beta_{HA} ~=~ (\alpha_{HA}) * \frac{1}{P_{(black|pelagic)ayu}^{HA} - 1}, \end{equation}\]

\[\begin{equation} var_{P_{HA}} ~=~ (P_{(black|pelagic)ayu}^{HA} * cvP_{(black|pelagic)ayu}^{HA})^2 \end{equation}\]

where \(cvP_{(black|pelagic)ayu}^{HA}\) is the coefficient of variation for the hydroacoustic proportions

\[\begin{equation} cvP_{(black|pelagic)ayu}^{HA} ~=~ \frac{\sqrt{varP_{(black|pelagic)ayu}^{HA}}}{P_{(black|pelagic)ayu}^{HA}} \end{equation}\]

and the variance is approximated using the XXXX method as

\[\begin{equation} varP_{(black|pelagic)ayu}^{HA} ~=~ (\frac{1}{n_{pel}})^2 * varN_{black} + (\frac{n_{black}}{n_{pel}^2}) * varN_{pel} \end{equation}\]

where \(varN_{black}\) and \(varN_{black}\) are the variance of the estimated number of black and pelagic rockfish in the hydroacoustic survey, respectively (CITATION).

The average weight of rockfish by species, user, area and year was modeled hierarchically at several levels within regions such that

\[\begin{equation} wt_{(comp)ayu} ~\sim~ Normal(wt_{(comp)au},\sigma_{wt_{(comp)au}}) ~\sim~ Normal(wt_{(comp)a},\sigma_{wt_{(comp)a}}) ~\sim~ Normal(wt_{(comp)region},\sigma_{wt_{(comp)region}}) \end{equation}\]

where region refers to Kodiak, Southcentral and Southeast. Mean weights and variance were calculated as XXX.

Alternative likelihoods for release estimates

To loosen the assumption that logbook release data are an effective census of true releases I explored models that treated logbook release estimates as a lower bound on the estimate of true releases. In a hybrid approach yelloweye and non-pelagic releases are regarded as a reliable census (given the emphasis and ease of recording these fish) but censors the pelagic and total rockfish release estimates (where censoring implies NA values) such that

\[\begin{equation} \text{censored} \widehat{LB}_R{ay}~\sim~\textrm{LogNormal}\left(\log(R_{ay}), 1\right)\text{T}\left(\widehat{LB}_R{ay}, \infty\right) \\ \text{censored} \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{LogNormal}\left(\log(R_{(pelagic)ay}), 1\right)\text{T}\left(\widehat{LB}_R{(pelagic)ay}, \infty\right) \\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

This model formulation failed such that there was not enough data to inform pelagic releases and the values did not seem valid. A second approach is being explored that fits the censored data using a lognormal distribution centered around the logbook release value, but also with a lower bound equal to the number of recorded releases such that

\[\begin{equation} \widehat{LB}_R{ay}~\sim~\textrm{LogNormal}\left(\log(R_{ay}), \sigma_{Ray1}^2\right)\text{T}\left(\widehat{LB}_R{ay}, \infty\right) \\ \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{LogNormal}\left(\log(R_{(pelagic)ay}), \sigma_{Ray1}^2\right)\text{T}\left(\widehat{LB}_R{(pelagic)ay}, \infty\right) \\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

Logbook data is assumed to be a census and as such there is no estimate of uncertainty. As of this writing, several methods are being examined for how to treat \(\sigma_{Ray1}^2\). Models are being run that attempt to allow the model to estimate \(\sigma_{Ray1}^2\) with priors. A simple model applies a uniform prior (0.1,50) to \(\sigma_{Ray1}^2\). A hierarchichal approach based on regions is also being examined whereby \(\sigma_{Ray1}^2\) is lognormally distributed around hyper priors \(\mu_{\sigma_R}\) and \(\sigma_{\sigma_R}\). Initial efforts have applied a uniform prior on \(\mu_{\sigma_R}\) between 1 and 50 and on \(\sigma_{\sigma_R}\) between 0 and 10.

Priors.

Priors range from uninformative to very informative or fixed. Priors for compositional logistic parameters are in Table 2 and proportion harvest logistic parameters are in Table 3. Until I figure out how to make a nice table in Rmarkdown, please refer to the attached spreadsheet and comp and harvp tabs.

Unresolved issues and outstanding questions:

  1. Reliability of unguided release estimates: These estimates have the least information feeding them and rely on the bias-corrected SWHS release estimates of all rockfish and the trends in release probability evident in the logbook data. The \(\beta4\) term that estimates the guided/unguided effect was given a very informative prior that tied the release probability of private anglers tightly to that of the charter fleet. The model is then trying to balance the three species complex estimates (pelagic, yelloweye and other) so that they sum to the total unguided releases estimated from the bias corrected SWHS data. For the most part this seems reasonable and appears to work, but there are certain areas where the estimates are “wonky”:

    1. Total rockfish releases more or less align with the total releases estimated with the Howard methods. Presumably, much of the discrepancy results from the substantial bias in release estimates from the SWHS. Interestingly, the logbook data indicates that the SWHS underestimates harvests but overestimates releases by a significant factor (Figure 23 and 24 below).
    2. In general, release estimates of black rockfish are substantially lower than those calculated using the Howard methods. Presumably, much of this derives from the bias correction of the SWHS release estimates.
    3. Yelloweye release estimates also differ considerably from the Howard estimates, but unlike black rockfish are sometimes lower and sometimes higher. Two areas in particular are a little head scratching. Yelloweye releases in the Kodiak Northeast area in particular are significantly lower than for guided anglers with the same pattern evident in Cook Inlet to a lesser extent. Cook Inlet yelloweye numbers are very small, so this is a sample size issue with little consequence. The cause of the Kodiak northeast estimates is not clear to me at this point, but the model estimates the proportion harvested by unguided anglers to be much lower than that of guided anglers, even with the informative prior on \(\beta4\). This must be a product of the bias corrected SWHS release estimates and how the model is partitioning that estimate into the 3 species complexes, but itis a bit a of head scratcher.
  2. Proportion guided estimates: There is not much data on this proportion prior to 2011 and it is not modeled with any sort of trend as was done for species composition and harvest proportions. With the exception of Cook Inlet and North Gulf Coast areas, there is little, if any, trend apparent in the data and perhaps this approach is the best available given the data available. However, if there are data sources somewhere that could inform this part of the model they could be incorporated.

  3. Prior choices in general need to be vetted. The priors on the logistic curves are fairly informed in an effort to achieve the desired shapes for hindcasting. Ideally, sensitivity testing would occur but the model is very slow to converge. The beta parameters on the logistic curves have required a lot of work on the priors to reach convergence.

  4. Proportion harvest estimates for non-pelagic, non-yelloweye in Kodiak WKMA: I need to adjust the prior on the inflection point, \(\beta3\), so that it is forced to occur after 2006. Right now the model is estimating inflection in two Kodiak areas before that point where there is no data to justify a shift. The current inflection is a result of the hierachichal model.

  5. Proportion pelagic in PWS and CSEO: The parameters for these particular proportions are very slow to converge. For the CSEO, the estimates of the \(\beta\) parameters are similar to the other Southeast areas, but the mixing is poor over the length of the chains. In this case I think they will ultimately converge with a very long model run and the shape of the curve in the model output looks acceptable. For the two PWS areas the model seems to struggle with the disparate proportional data from the logbook and the port sampling. There is some wandering in the chains of the \(\beta0\) and \(\beta1\) terms and spikiness in the \(\beta2\) terms. I’ve been working on constraining the hyperpriors for PWS \(beta2\). Similar to CSEO, it may just entail a very long model run to reach convergence, but the shape of the curves looks reasonable.

Next steps:

Once the model is finalized, harvest and release numbers need to be converted into biomass removals. This is a two step process where release mortality estimates are applied to the release estimates to estimate the number of released rockfish that do not survive. This is based on studies and will reflect the values that the department has been using with the Howard methods. Region 2 (both Southcentral and Kodiak) have release-at-depth estimates from a number of years that they apply across all years and then calculate mortality rates based on those estiates. Southease does not have release-at-depth data and simply applies an assumed rate based on research.

Once release mortality is calculated average weight data is applied to convert numbers to biomass. The plan is to incorporate all of this into the model to propogate uncertainty into the posteriors. However, the model already takes a long time to run and I may explore a simpler approach using the posteriors from the numbers model to speed up processing.

Results

**Figure X.**- Rhat values and proportion of parameters that converged (Rhat < 1.1.)

Figure X.- Rhat values and proportion of parameters that converged (Rhat < 1.1.)

Estimate comparison

Since previous estimates of rockfish harvest have been produced these first 3 graphs will be used to show how the modeled estimates compare to the estimates produced earlier. For total rockfish the estimates are in general agreement although differences are noted. These estimates should be more reliable because they include both SWHS and guide logbook data, handle variance more appropriately, use hierarchical distributions when data is missing, directly consider observation error and are produced using reproducible research.

**Figure 3.**- Total rockfish harvests 1996-2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 3.- Total rockfish harvests 1996-2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.



**Figure 3.**- Total rockfish releases 1996--2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 3.- Total rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.


Notes from Adam: When looking at only black rockfish the most significant differences are for the Prince William Sound Inside area. I did not spend a great deal of time tracking this down although it looks like the previous version used bad values for \(P_{(black)ayu}\) for at least unguided anglers. For the moment I would ignore the results for BSIA and SOKO2SAP. I think it is possible to give approximate values for these areas but it will require a little more coding which I have yet to do.

**Figure 4.**- Black rockfish harvests 1996--2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 4.- Black rockfish harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.


And black rockfish releases…

**Figure 5.**- Black rockfish releases 1996--2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 5.- Black rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.





**Figure 6.**- Yellow rockfish harvests 1996--2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 6.- Yellow rockfish harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.



**Figure 7.**- Yellow rockfish releases 1996--2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 7.- Yellow rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.





**Figure 8.**- DSR rockfish (including yelloweye) harvests 1996--2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 8.- DSR rockfish (including yelloweye) harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.



**Figure 9.**- DSR rockfish releases (including yelloweye) 1996--2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 9.- DSR rockfish releases (including yelloweye) 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.



**Figure 11.**- Slope rockfish harvests 1996-2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 11.- Slope rockfish harvests 1996-2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.



**Figure 12.**- Slope rockfish releases 1996--2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 12.- Slope rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Total Biomass Removal Estimates

**Figure 13.**- Black rockfish estimated total removals in lbs in 1996--2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 13.- Black rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

**Figure 14.**- Yellow rockfish estimated total removals in lbs in 1996--2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 14.- Yellow rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

**Figure 15.**- Pelagic rockfish estimated total removals in lbs in 1996--2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 15.- Pelagic rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.


**Figure 16.**- Non-yelloweye, demersal shelf rockfish estimated total removals in lbs in 1996-2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 16.- Non-yelloweye, demersal shelf rockfish estimated total removals in lbs in 1996-2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.


**Figure 17.**- Slope rockfish estimated total removals in lbs in 1996-2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 17.- Slope rockfish estimated total removals in lbs in 1996-2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.


Model fit

Logbook residuals

**Figure 18.**- Residuals from logbook harvests.

Figure 18.- Residuals from logbook harvests.


SWHS residuals

**Figure 19.**- Residuals from SWHS harvests.

Figure 19.- Residuals from SWHS harvests.



**Figure 20.**- Residual of SWHS releases.

Figure 20.- Residual of SWHS releases.

Parameter estimates

P(Charter)

These histograms show the posterior distribution of the mean percent of rockfish harvested by the charter fleet.

**Figure 21.**- Mean percent of harvest by charter anglers.

Figure 21.- Mean percent of harvest by charter anglers.


When considered annually we see the percent of rockfish harvested by the charter fleet follows our data fairly well although the model smooths out the changes and we just do not have much information about this ratio. Prior to 2011 the percent charter is confounded with SWHS bias and should be mostly discounted.

**Figure 22.**- Annual estimates of the percent of harvest by charter anglers for 16 commerical fishing manamgent areas, 1996-2023.

Figure 22.- Annual estimates of the percent of harvest by charter anglers for 16 commerical fishing manamgent areas, 1996-2023.

P(Harvest)

These plots show the fitted logistic line to the proportion of caught rockfish that are harvested. These estimates are used for hindcasting catch estimates based on the harvest data in early years when catch estimates are unavailable.


**Figure 23.**- Annual proportion of pelagic rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available.

Figure 23.- Annual proportion of pelagic rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available.


**Figure 24.**- Annual proportion of yelloweye rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available.

Figure 24.- Annual proportion of yelloweye rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available.


**Figure 25.**- Annual proportion of non-pelagic, non-yelloweye rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available. Note, that this is not estimated for Southeast areas because non=pelagics are divided between DSR (including yelloweye) and Slope species.

Figure 25.- Annual proportion of non-pelagic, non-yelloweye rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available. Note, that this is not estimated for Southeast areas because non=pelagics are divided between DSR (including yelloweye) and Slope species.


## NULL


## NULL

SWHS bias

Figure 23 shows the mean estimate for SWHS bias in harvests and releases. Cook Inlet, North Gulf Coast and North Southeast Inside all look pretty good while most other areas have substantial bias. Prince William Sound Inside has the largest bias. Bias in release estimates is substantial and whereas the SWHS appears to underestimate harvests, it appears to greatly overestimates releases by a factor of 2 or more in most areas as derived from logbook reported releases.

**Figure 28.**- Mean SWHS bias for harvests and catches. Note that a bias < 1 indicates that the SWHS *underestimates* the true value and bias > 1 indicates the survey *overestimates* the true value.

Figure 28.- Mean SWHS bias for harvests and catches. Note that a bias < 1 indicates that the SWHS underestimates the true value and bias > 1 indicates the survey overestimates the true value.


Our estimates of SWHS harvest bias track observations fairly well when he have guided harvest estimates. The estimates of release bias in the SWHS data track observed patterns to an extent, but appear to smooth these more volatile disagreements with the logbook data. Adam postulated in his initial start on this that some of this could be the result of the estimates of the proportion guided. This value was not modelled with a trend and thus applies a constant estimate when hindcasting. Data on these relationships could greatly improve this model.

**Figure 29.**- Annual estimates of SWHS bias in harvests and releases for 16 commerical fishing manamgent areas, 1996-2023. Note that a bias < 1 indicates that the SWHS *underestimates* the true value and bias > 1 indicates the survey *overestimates* the true value.

Figure 29.- Annual estimates of SWHS bias in harvests and releases for 16 commerical fishing manamgent areas, 1996-2023. Note that a bias < 1 indicates that the SWHS underestimates the true value and bias > 1 indicates the survey overestimates the true value.

P(pelagic)

We model the percentage of pelagic rockfish in the harvest because we have the information for charter anglers (via logbooks) starting in 1998. Other than looking at the model estimates you can use Figure 25 to compare the two data streams for pelagic rockfish harvest. In general they are in agreement with major exceptions in Price William Sound inside, Prince William Sound outside (early in the time series) and South Southeast inside.

**Figure 30.**- Annual estimates of the percent of the sport harvest that was pelagic rockfish for 16 commerical fishing manamgent areas, 1996-2023.

Figure 30.- Annual estimates of the percent of the sport harvest that was pelagic rockfish for 16 commerical fishing manamgent areas, 1996-2023.

P(black|pelagic)

Note that in Southeast Alaska we only have composition data starting in 2006. Tania dug up old SE data, but it did not provide any useful data for species apportionment. For the most part, P(black|pelagic) is relatively constant across areas, with the exception of Cook Inlet and NSEI in Southeast AK. It may be worth discussing whether the shifts in those areas is a result of improved or changing species identification rather than actual shift in the species composition of the catch.

**Figure 31.**- Annual estimates of the percent of the sport harvest of pelagic rockfish that were black rockfish for 16 commerical fishing manamgent areas, 1996-2023. Kodiak panels include data from a hydroacoustic survey and the proportion of pelagic rockfish that are black in those areas (red) and the adjusted proportions based on obseved harvests for charter (blue) and private (cyan) users.

Figure 31.- Annual estimates of the percent of the sport harvest of pelagic rockfish that were black rockfish for 16 commerical fishing manamgent areas, 1996-2023. Kodiak panels include data from a hydroacoustic survey and the proportion of pelagic rockfish that are black in those areas (red) and the adjusted proportions based on obseved harvests for charter (blue) and private (cyan) users.

P(yelloweye|non-pelagic / yelloweye|DSR)

**Figure 32.**- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were yelloweye rockfish for 16 commerical fishing manamgent areas, 1996-2023. Note that P(yelloweye) is the the proportion relative to non-pelagics for Central and Kodiak areas but is relative to DSR for Southeast areas.

Figure 32.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were yelloweye rockfish for 16 commerical fishing manamgent areas, 1996-2023. Note that P(yelloweye) is the the proportion relative to non-pelagics for Central and Kodiak areas but is relative to DSR for Southeast areas.

P(DSR|non-pelagic)

**Figure 33.**- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were DSR rockfish for 6 Southeast commerical fishing manamgent areas, 1996-2023.

Figure 33.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were DSR rockfish for 6 Southeast commerical fishing manamgent areas, 1996-2023.

P(slope|non-pelagic)

**Figure 34.**- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were slope rockfish for 6 southeast commerical fishing manamgent areas, 1996-2023. Note that P(yelloweye) is the the proportion relative to non-pelagics for Central and Kodiak areas but is relative to DSR for Southeast areas.

Figure 34.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were slope rockfish for 6 southeast commerical fishing manamgent areas, 1996-2023. Note that P(yelloweye) is the the proportion relative to non-pelagics for Central and Kodiak areas but is relative to DSR for Southeast areas.



P(slope|non-pelagic & non-yellowye) For release estimates

**Figure 35.**- Annual estimates of the percent of the sport non-pelagic, non-yelloweye releases that were slope rockfish for 6 southeast commerical fishing manamgent areas, 1996-2023.

Figure 35.- Annual estimates of the percent of the sport non-pelagic, non-yelloweye releases that were slope rockfish for 6 southeast commerical fishing manamgent areas, 1996-2023.



Weight Fits

**Figure 36.**- Mean weights of black rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 36.- Mean weights of black rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.


**Figure 37.**- Mean weights of yelloweye rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 37.- Mean weights of yelloweye rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.


**Figure 38.**- Mean weights of non-black, pelagic rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 38.- Mean weights of non-black, pelagic rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.


**Figure 39.**- Mean weights of non-yelloweye, demersal shelf rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 39.- Mean weights of non-yelloweye, demersal shelf rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.


**Figure 40.**- Mean weights of slope rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 40.- Mean weights of slope rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.


### Summary of unconverged parameters:

Table 1. Summary of unconverged parameters including the number (n) and the average Rhat from the unconverged parameters.
parameter n badRhat_avg
beta1_pelagic 3 1.399997
beta0_pelagic 3 1.377373
beta2_pelagic 2 1.353275
beta4_yellow 4 1.204574
beta4_pelagic 4 1.191563
parameter n badRhat_avg
beta1_pH 4 1.190083
beta2_pH 1 1.183916
beta3_pelagic 1 1.169991
beta3_yellow 1 1.145337
beta2_yellow 1 1.131006
Table 2. Summary of unconverged parameters by area
BSAI CSEO eastside NSEI NSEO PWSI PWSO SOKO2SAP WKMA
beta0_pelagic 0 1 0 0 0 1 1 0 0
beta1_pelagic 0 1 0 0 0 1 1 0 0
beta1_pH 1 0 0 0 0 1 1 1 0
beta2_pelagic 0 1 0 0 1 0 0 0 0
beta2_pH 0 0 0 0 0 0 0 0 1
beta2_yellow 0 0 0 1 0 0 0 0 0
beta3_pelagic 0 1 0 0 0 0 0 0 0
beta3_yellow 0 0 0 0 1 0 0 0 0
beta4_pelagic 0 1 1 0 0 0 0 1 1
beta4_yellow 1 0 1 0 0 0 0 1 1

Parameter estimates:

Summary Table of Parameter Estimates
Parameter mean sd Lower_CI Median Upper_CI
mu_bc_H[1] -0.128 0.070 -0.256 -0.131 0.017
mu_bc_H[2] -0.096 0.045 -0.173 -0.100 0.003
mu_bc_H[3] -0.433 0.072 -0.570 -0.435 -0.285
mu_bc_H[4] -0.984 0.194 -1.370 -0.981 -0.609
mu_bc_H[5] 0.921 0.901 -0.152 0.743 3.128
mu_bc_H[6] -2.163 0.324 -2.788 -2.173 -1.524
mu_bc_H[7] -0.457 0.108 -0.678 -0.455 -0.253
mu_bc_H[8] 0.241 0.364 -0.357 0.206 1.049
mu_bc_H[9] -0.294 0.132 -0.553 -0.292 -0.031
mu_bc_H[10] -0.107 0.070 -0.237 -0.109 0.038
mu_bc_H[11] -0.123 0.037 -0.194 -0.123 -0.050
mu_bc_H[12] -0.254 0.107 -0.477 -0.249 -0.059
mu_bc_H[13] -0.135 0.079 -0.288 -0.136 0.025
mu_bc_H[14] -0.299 0.098 -0.496 -0.300 -0.122
mu_bc_H[15] -0.345 0.049 -0.441 -0.346 -0.249
mu_bc_H[16] -0.260 0.386 -0.911 -0.292 0.540
mu_bc_R[1] 1.296 0.143 1.031 1.291 1.593
mu_bc_R[2] 1.453 0.093 1.272 1.454 1.638
mu_bc_R[3] 1.391 0.144 1.102 1.393 1.674
mu_bc_R[4] 0.917 0.207 0.482 0.929 1.303
mu_bc_R[5] 1.175 0.471 0.244 1.186 2.056
mu_bc_R[6] -1.596 0.422 -2.427 -1.597 -0.754
mu_bc_R[7] 0.461 0.212 0.020 0.474 0.860
mu_bc_R[8] 0.558 0.200 0.154 0.563 0.936
mu_bc_R[9] 0.347 0.203 -0.072 0.358 0.700
mu_bc_R[10] 1.295 0.135 1.011 1.299 1.558
mu_bc_R[11] 1.036 0.098 0.844 1.034 1.234
mu_bc_R[12] 0.822 0.202 0.431 0.819 1.220
mu_bc_R[13] 1.028 0.104 0.820 1.026 1.230
mu_bc_R[14] 0.894 0.143 0.619 0.895 1.171
mu_bc_R[15] 0.783 0.109 0.566 0.783 1.000
mu_bc_R[16] 1.090 0.127 0.840 1.093 1.343
tau_pH[1] 5.165 0.445 4.340 5.140 6.103
tau_pH[2] 2.031 0.226 1.609 2.024 2.502
tau_pH[3] 2.126 0.217 1.729 2.115 2.572
beta0_pH[1,1] 0.547 0.177 0.189 0.554 0.886
beta0_pH[2,1] 1.377 0.172 1.030 1.380 1.711
beta0_pH[3,1] 1.429 0.192 1.002 1.444 1.762
beta0_pH[4,1] 1.564 0.224 1.066 1.583 1.954
beta0_pH[5,1] -0.865 0.293 -1.472 -0.846 -0.350
beta0_pH[6,1] -0.717 0.388 -1.619 -0.660 -0.088
beta0_pH[7,1] -0.523 0.450 -1.475 -0.493 0.351
beta0_pH[8,1] -0.650 0.275 -1.251 -0.628 -0.181
beta0_pH[9,1] -0.666 0.288 -1.279 -0.633 -0.177
beta0_pH[10,1] 0.229 0.193 -0.150 0.232 0.594
beta0_pH[11,1] -0.078 0.162 -0.406 -0.072 0.228
beta0_pH[12,1] 0.496 0.187 0.128 0.495 0.843
beta0_pH[13,1] 0.015 0.155 -0.301 0.018 0.312
beta0_pH[14,1] -0.318 0.169 -0.672 -0.311 -0.006
beta0_pH[15,1] -0.028 0.182 -0.394 -0.022 0.308
beta0_pH[16,1] -0.484 0.361 -1.385 -0.423 0.052
beta0_pH[1,2] 2.847 0.165 2.508 2.855 3.157
beta0_pH[2,2] 2.890 0.134 2.624 2.889 3.146
beta0_pH[3,2] 3.136 0.149 2.860 3.132 3.448
beta0_pH[4,2] 2.952 0.133 2.683 2.949 3.218
beta0_pH[5,2] 4.878 1.498 2.971 4.550 8.654
beta0_pH[6,2] 3.112 0.203 2.715 3.110 3.530
beta0_pH[7,2] 1.837 0.195 1.453 1.836 2.228
beta0_pH[8,2] 2.866 0.172 2.537 2.871 3.196
beta0_pH[9,2] 3.443 0.223 3.011 3.441 3.883
beta0_pH[10,2] 3.753 0.201 3.364 3.755 4.148
beta0_pH[11,2] -4.827 0.294 -5.418 -4.823 -4.255
beta0_pH[12,2] -4.779 0.400 -5.607 -4.764 -4.009
beta0_pH[13,2] -4.566 0.390 -5.320 -4.581 -3.759
beta0_pH[14,2] -5.595 0.477 -6.554 -5.582 -4.727
beta0_pH[15,2] -4.284 0.329 -4.918 -4.276 -3.630
beta0_pH[16,2] -4.839 0.374 -5.634 -4.833 -4.120
beta0_pH[1,3] -0.142 0.716 -1.722 -0.057 1.016
beta0_pH[2,3] 2.185 0.162 1.867 2.183 2.513
beta0_pH[3,3] 2.529 0.151 2.244 2.530 2.832
beta0_pH[4,3] 2.960 0.163 2.630 2.963 3.276
beta0_pH[5,3] 2.166 1.436 0.399 1.853 5.755
beta0_pH[6,3] 1.001 0.489 -0.151 1.034 1.877
beta0_pH[7,3] 0.625 0.173 0.302 0.620 0.967
beta0_pH[8,3] 0.307 0.187 -0.059 0.311 0.659
beta0_pH[9,3] -0.624 0.372 -1.512 -0.593 0.012
beta0_pH[10,3] 0.474 0.375 -0.409 0.507 1.095
beta0_pH[11,3] -0.165 0.323 -0.760 -0.180 0.511
beta0_pH[12,3] -0.852 0.348 -1.584 -0.827 -0.248
beta0_pH[13,3] -0.125 0.311 -0.716 -0.131 0.491
beta0_pH[14,3] -0.272 0.262 -0.779 -0.276 0.255
beta0_pH[15,3] -0.688 0.302 -1.285 -0.683 -0.110
beta0_pH[16,3] -0.391 0.294 -0.956 -0.390 0.216
beta1_pH[1,1] 3.080 0.313 2.530 3.049 3.743
beta1_pH[2,1] 2.140 0.263 1.671 2.126 2.688
beta1_pH[3,1] 1.973 0.323 1.453 1.940 2.672
beta1_pH[4,1] 2.381 0.346 1.830 2.330 3.195
beta1_pH[5,1] 2.303 0.364 1.709 2.281 3.034
beta1_pH[6,1] 3.747 0.970 2.361 3.563 6.031
beta1_pH[7,1] 2.633 0.874 0.981 2.567 4.514
beta1_pH[8,1] 4.004 0.999 2.644 3.768 6.537
beta1_pH[9,1] 2.375 0.422 1.724 2.321 3.331
beta1_pH[10,1] 2.397 0.270 1.905 2.390 2.962
beta1_pH[11,1] 3.258 0.208 2.862 3.256 3.667
beta1_pH[12,1] 2.536 0.218 2.130 2.532 2.967
beta1_pH[13,1] 2.961 0.223 2.542 2.946 3.428
beta1_pH[14,1] 3.421 0.220 3.017 3.414 3.886
beta1_pH[15,1] 2.528 0.230 2.099 2.523 2.999
beta1_pH[16,1] 4.124 0.643 3.210 4.018 5.738
beta1_pH[1,2] 0.000 0.000 0.000 0.000 0.000
beta1_pH[2,2] 0.000 0.000 0.000 0.000 0.000
beta1_pH[3,2] 0.000 0.004 0.000 0.000 0.001
beta1_pH[4,2] 0.000 0.011 0.000 0.000 0.001
beta1_pH[5,2] 0.000 0.000 0.000 0.000 0.000
beta1_pH[6,2] 0.000 0.000 0.000 0.000 0.000
beta1_pH[7,2] 0.000 0.000 0.000 0.000 0.000
beta1_pH[8,2] 0.000 0.000 0.000 0.000 0.000
beta1_pH[9,2] 0.000 0.000 0.000 0.000 0.000
beta1_pH[10,2] 0.000 0.000 0.000 0.000 0.000
beta1_pH[11,2] 6.665 0.325 6.024 6.666 7.304
beta1_pH[12,2] 6.449 0.466 5.574 6.437 7.448
beta1_pH[13,2] 6.948 0.433 6.074 6.952 7.788
beta1_pH[14,2] 7.237 0.497 6.332 7.232 8.242
beta1_pH[15,2] 6.764 0.365 6.046 6.765 7.492
beta1_pH[16,2] 7.432 0.417 6.655 7.415 8.296
beta1_pH[1,3] 4.689 1.651 2.140 4.444 8.075
beta1_pH[2,3] 0.000 0.000 0.000 0.000 0.000
beta1_pH[3,3] 0.000 0.000 0.000 0.000 0.000
beta1_pH[4,3] 0.000 0.000 0.000 0.000 0.000
beta1_pH[5,3] 3.695 5.628 0.852 2.774 12.240
beta1_pH[6,3] 3.204 5.772 0.402 2.613 8.008
beta1_pH[7,3] 0.000 0.000 0.000 0.000 0.000
beta1_pH[8,3] 2.746 0.343 2.067 2.740 3.426
beta1_pH[9,3] 2.753 0.447 2.016 2.724 3.768
beta1_pH[10,3] 2.904 0.441 2.148 2.865 3.914
beta1_pH[11,3] 2.745 0.383 1.980 2.751 3.505
beta1_pH[12,3] 4.097 0.435 3.286 4.074 4.971
beta1_pH[13,3] 1.705 0.339 1.032 1.713 2.342
beta1_pH[14,3] 2.517 0.333 1.853 2.519 3.165
beta1_pH[15,3] 1.976 0.331 1.296 1.977 2.615
beta1_pH[16,3] 1.800 0.323 1.143 1.796 2.433
beta2_pH[1,1] 0.482 0.123 0.289 0.467 0.762
beta2_pH[2,1] 0.592 0.307 0.262 0.522 1.364
beta2_pH[3,1] 0.671 0.493 0.212 0.562 1.886
beta2_pH[4,1] 0.484 0.205 0.209 0.448 0.951
beta2_pH[5,1] 1.441 0.956 0.244 1.317 3.663
beta2_pH[6,1] 0.187 0.062 0.099 0.178 0.333
beta2_pH[7,1] 0.027 0.292 0.000 0.000 0.143
beta2_pH[8,1] 0.247 0.108 0.129 0.229 0.464
beta2_pH[9,1] 0.418 0.205 0.164 0.383 0.876
beta2_pH[10,1] 0.614 0.283 0.294 0.558 1.290
beta2_pH[11,1] 0.785 0.200 0.482 0.752 1.241
beta2_pH[12,1] 1.381 0.475 0.750 1.285 2.629
beta2_pH[13,1] 0.746 0.235 0.408 0.708 1.292
beta2_pH[14,1] 0.848 0.223 0.529 0.812 1.405
beta2_pH[15,1] 0.819 0.303 0.415 0.757 1.552
beta2_pH[16,1] 0.373 0.170 0.171 0.326 0.839
beta2_pH[1,2] 0.000 0.000 0.000 0.000 0.000
beta2_pH[2,2] 0.000 0.000 0.000 0.000 0.000
beta2_pH[3,2] -1.995 1.835 -6.798 -1.513 -0.030
beta2_pH[4,2] -2.015 1.868 -6.752 -1.499 -0.026
beta2_pH[5,2] 0.000 0.000 0.000 0.000 0.000
beta2_pH[6,2] 0.000 0.000 0.000 0.000 0.000
beta2_pH[7,2] 0.000 0.000 0.000 0.000 0.000
beta2_pH[8,2] 0.000 0.000 0.000 0.000 0.000
beta2_pH[9,2] 0.000 0.000 0.000 0.000 0.000
beta2_pH[10,2] 0.000 0.000 0.000 0.000 0.000
beta2_pH[11,2] -9.663 4.407 -20.699 -8.581 -4.116
beta2_pH[12,2] -8.012 5.133 -20.194 -7.102 -0.906
beta2_pH[13,2] -7.910 5.067 -20.265 -6.805 -1.730
beta2_pH[14,2] -8.521 4.700 -20.356 -7.376 -2.512
beta2_pH[15,2] -9.397 4.490 -20.668 -8.282 -3.721
beta2_pH[16,2] -9.598 4.460 -20.695 -8.617 -3.910
beta2_pH[1,3] 0.242 0.261 0.101 0.181 0.700
beta2_pH[2,3] 0.000 0.000 0.000 0.000 0.000
beta2_pH[3,3] 0.000 0.000 0.000 0.000 0.000
beta2_pH[4,3] 0.000 0.000 0.000 0.000 0.000
beta2_pH[5,3] 9.189 6.428 -0.087 8.337 23.912
beta2_pH[6,3] 9.243 6.264 0.268 8.216 23.770
beta2_pH[7,3] 0.000 0.000 0.000 0.000 0.000
beta2_pH[8,3] 10.280 5.834 1.948 9.139 24.335
beta2_pH[9,3] 9.120 6.280 0.520 7.994 23.946
beta2_pH[10,3] 8.682 6.561 0.497 7.618 23.867
beta2_pH[11,3] -2.211 1.973 -8.025 -1.634 -0.610
beta2_pH[12,3] -2.396 1.863 -7.703 -1.862 -0.940
beta2_pH[13,3] -2.826 2.266 -9.655 -2.083 -0.792
beta2_pH[14,3] -2.759 2.134 -8.785 -2.071 -0.880
beta2_pH[15,3] -2.911 2.240 -9.600 -2.168 -1.003
beta2_pH[16,3] -2.979 2.294 -9.592 -2.216 -0.892
beta3_pH[1,1] 35.918 0.810 34.435 35.883 37.606
beta3_pH[2,1] 33.571 1.109 31.654 33.476 36.046
beta3_pH[3,1] 33.650 1.133 31.669 33.600 35.969
beta3_pH[4,1] 33.754 1.148 31.702 33.751 36.150
beta3_pH[5,1] 27.681 1.074 26.424 27.463 30.821
beta3_pH[6,1] 37.967 3.092 32.510 37.760 44.660
beta3_pH[7,1] 30.604 7.961 18.486 30.169 44.909
beta3_pH[8,1] 40.007 2.153 36.413 39.716 45.025
beta3_pH[9,1] 30.761 1.759 28.071 30.602 34.161
beta3_pH[10,1] 32.716 0.887 31.075 32.677 34.569
beta3_pH[11,1] 30.349 0.468 29.450 30.355 31.303
beta3_pH[12,1] 30.168 0.393 29.368 30.168 30.915
beta3_pH[13,1] 33.183 0.597 32.041 33.175 34.438
beta3_pH[14,1] 32.024 0.463 31.156 32.021 32.971
beta3_pH[15,1] 31.175 0.649 29.925 31.161 32.464
beta3_pH[16,1] 32.037 1.080 30.334 31.899 34.544
beta3_pH[1,2] 30.207 7.947 18.410 29.485 44.756
beta3_pH[2,2] 30.061 7.859 18.487 29.178 44.683
beta3_pH[3,2] 29.928 7.944 18.506 28.934 44.876
beta3_pH[4,2] 29.944 7.885 18.504 28.979 44.880
beta3_pH[5,2] 29.816 7.948 18.533 28.722 44.968
beta3_pH[6,2] 29.944 7.948 18.504 29.047 44.881
beta3_pH[7,2] 29.856 7.939 18.521 28.869 44.826
beta3_pH[8,2] 29.803 7.946 18.474 28.512 44.986
beta3_pH[9,2] 29.960 7.920 18.482 28.959 44.747
beta3_pH[10,2] 29.882 7.954 18.487 28.722 45.075
beta3_pH[11,2] 43.407 0.179 43.119 43.390 43.772
beta3_pH[12,2] 43.189 0.198 42.924 43.141 43.687
beta3_pH[13,2] 43.865 0.146 43.480 43.904 44.045
beta3_pH[14,2] 43.305 0.204 43.045 43.252 43.808
beta3_pH[15,2] 43.416 0.195 43.111 43.395 43.806
beta3_pH[16,2] 43.491 0.187 43.154 43.489 43.837
beta3_pH[1,3] 39.113 3.184 32.997 39.041 45.320
beta3_pH[2,3] 30.506 7.950 18.506 29.722 45.095
beta3_pH[3,3] 30.184 7.863 18.446 29.225 44.970
beta3_pH[4,3] 30.291 8.132 18.345 29.711 45.042
beta3_pH[5,3] 36.844 3.872 31.251 36.238 45.043
beta3_pH[6,3] 40.499 3.496 31.763 40.959 45.615
beta3_pH[7,3] 37.937 4.290 31.359 37.575 45.519
beta3_pH[8,3] 41.488 0.248 41.048 41.487 41.933
beta3_pH[9,3] 33.497 0.540 31.874 33.583 34.233
beta3_pH[10,3] 35.848 0.785 33.440 36.022 36.840
beta3_pH[11,3] 41.801 0.827 39.988 41.856 43.252
beta3_pH[12,3] 41.728 0.389 40.962 41.742 42.497
beta3_pH[13,3] 42.756 0.896 41.096 42.769 44.712
beta3_pH[14,3] 41.091 0.584 39.834 41.126 42.181
beta3_pH[15,3] 42.549 0.702 41.032 42.633 43.713
beta3_pH[16,3] 42.894 0.757 41.143 43.021 44.088
beta0_pelagic[1] 2.224 0.132 1.974 2.219 2.488
beta0_pelagic[2] 1.518 0.128 1.274 1.517 1.767
beta0_pelagic[3] -0.609 0.769 -2.279 -0.504 0.500
beta0_pelagic[4] -0.511 0.859 -2.638 -0.310 0.724
beta0_pelagic[5] 1.185 0.253 0.686 1.190 1.688
beta0_pelagic[6] 1.458 0.265 0.913 1.472 1.942
beta0_pelagic[7] 1.590 0.207 1.203 1.580 2.040
beta0_pelagic[8] 1.768 0.204 1.385 1.760 2.196
beta0_pelagic[9] 2.481 0.315 1.850 2.489 3.048
beta0_pelagic[10] 2.527 0.207 2.081 2.540 2.905
beta0_pelagic[11] 0.106 0.486 -1.399 0.217 0.702
beta0_pelagic[12] 1.681 0.144 1.399 1.686 1.955
beta0_pelagic[13] 0.296 0.213 -0.189 0.314 0.657
beta0_pelagic[14] -0.097 0.269 -0.682 -0.073 0.357
beta0_pelagic[15] -0.262 0.141 -0.537 -0.264 0.021
beta0_pelagic[16] 0.275 0.290 -0.478 0.343 0.667
beta1_pelagic[1] 0.000 0.000 0.000 0.000 0.000
beta1_pelagic[2] 0.000 0.000 0.000 0.000 0.000
beta1_pelagic[3] 2.428 1.324 0.600 2.164 5.319
beta1_pelagic[4] 2.008 1.080 0.537 1.776 4.661
beta1_pelagic[5] -0.064 0.315 -0.707 -0.068 0.541
beta1_pelagic[6] -0.092 0.447 -0.875 -0.124 0.741
beta1_pelagic[7] -0.016 0.292 -0.585 -0.017 0.570
beta1_pelagic[8] -0.017 0.285 -0.564 -0.020 0.561
beta1_pelagic[9] 0.202 0.486 -0.758 0.317 0.969
beta1_pelagic[10] 0.073 0.265 -0.441 0.065 0.604
beta1_pelagic[11] 3.642 1.179 2.166 3.421 6.521
beta1_pelagic[12] 2.773 0.324 2.197 2.761 3.413
beta1_pelagic[13] 2.961 0.751 1.793 2.879 4.760
beta1_pelagic[14] 4.330 1.072 2.753 4.138 6.832
beta1_pelagic[15] 2.910 0.267 2.386 2.914 3.422
beta1_pelagic[16] 3.633 0.939 2.665 3.303 6.398
beta2_pelagic[1] 0.000 0.000 0.000 0.000 0.000
beta2_pelagic[2] 0.000 0.000 0.000 0.000 0.000
beta2_pelagic[3] 0.558 2.069 0.032 0.108 6.194
beta2_pelagic[4] 1.103 3.000 0.026 0.285 11.093
beta2_pelagic[5] -0.020 0.665 -1.406 -0.015 1.354
beta2_pelagic[6] -0.099 0.683 -1.487 -0.138 1.359
beta2_pelagic[7] -0.027 0.651 -1.414 -0.015 1.324
beta2_pelagic[8] -0.003 0.655 -1.348 -0.010 1.436
beta2_pelagic[9] 0.200 0.693 -1.302 0.257 1.572
beta2_pelagic[10] 0.032 0.620 -1.309 0.037 1.372
beta2_pelagic[11] 1.938 3.994 0.106 0.305 14.160
beta2_pelagic[12] 6.141 5.075 0.978 4.648 19.655
beta2_pelagic[13] 0.926 2.018 0.172 0.441 6.105
beta2_pelagic[14] 0.337 0.287 0.161 0.290 0.746
beta2_pelagic[15] 6.490 5.124 1.227 5.049 21.020
beta2_pelagic[16] 4.773 5.557 0.195 3.030 20.001
beta3_pelagic[1] 29.640 7.892 18.439 28.405 44.784
beta3_pelagic[2] 29.837 7.982 18.465 28.876 44.812
beta3_pelagic[3] 30.254 7.269 18.853 29.305 44.698
beta3_pelagic[4] 25.208 5.492 18.477 24.095 40.687
beta3_pelagic[5] 30.177 8.313 18.506 28.770 45.313
beta3_pelagic[6] 31.846 6.789 19.035 31.597 44.357
beta3_pelagic[7] 29.814 7.843 18.418 28.926 44.822
beta3_pelagic[8] 29.849 8.038 18.482 28.661 44.733
beta3_pelagic[9] 30.806 6.062 19.303 31.039 42.597
beta3_pelagic[10] 29.413 8.100 18.418 28.153 44.794
beta3_pelagic[11] 42.586 1.826 37.683 43.018 45.571
beta3_pelagic[12] 43.472 0.290 42.987 43.460 43.996
beta3_pelagic[13] 42.828 1.317 40.424 42.809 45.487
beta3_pelagic[14] 42.386 1.655 39.099 42.333 45.564
beta3_pelagic[15] 43.182 0.256 42.552 43.182 43.676
beta3_pelagic[16] 43.079 0.909 40.618 43.191 45.069
mu_beta0_pelagic[1] 0.580 1.098 -1.861 0.613 2.791
mu_beta0_pelagic[2] 1.809 0.390 0.977 1.813 2.559
mu_beta0_pelagic[3] 0.331 0.459 -0.568 0.347 1.236
tau_beta0_pelagic[1] 0.402 0.433 0.050 0.258 1.541
tau_beta0_pelagic[2] 2.616 2.798 0.247 1.918 9.242
tau_beta0_pelagic[3] 1.529 1.171 0.178 1.258 4.476
beta0_yellow[1] -0.539 0.197 -0.996 -0.522 -0.210
beta0_yellow[2] 0.498 0.170 0.137 0.507 0.796
beta0_yellow[3] -0.327 0.200 -0.756 -0.314 0.024
beta0_yellow[4] 0.833 0.297 0.107 0.886 1.213
beta0_yellow[5] -0.301 0.349 -0.993 -0.310 0.385
beta0_yellow[6] 1.115 0.166 0.790 1.115 1.450
beta0_yellow[7] 0.979 0.159 0.677 0.978 1.298
beta0_yellow[8] 1.012 0.152 0.710 1.013 1.309
beta0_yellow[9] 0.664 0.158 0.354 0.664 0.971
beta0_yellow[10] 0.578 0.142 0.303 0.576 0.863
beta0_yellow[11] -1.940 0.427 -2.834 -1.938 -1.135
beta0_yellow[12] -3.697 0.433 -4.651 -3.672 -2.919
beta0_yellow[13] -3.692 0.456 -4.656 -3.667 -2.873
beta0_yellow[14] -2.064 0.565 -3.038 -2.120 -0.541
beta0_yellow[15] -2.856 0.403 -3.710 -2.834 -2.113
beta0_yellow[16] -2.358 0.444 -3.211 -2.369 -1.437
beta1_yellow[1] 0.788 0.866 0.010 0.643 2.467
beta1_yellow[2] 1.077 0.373 0.584 1.019 2.040
beta1_yellow[3] 0.723 0.300 0.237 0.708 1.397
beta1_yellow[4] 1.404 0.811 0.654 1.184 3.960
beta1_yellow[5] 0.000 0.000 0.000 0.000 0.000
beta1_yellow[6] 0.000 0.000 0.000 0.000 0.000
beta1_yellow[7] 0.000 0.000 0.000 0.000 0.000
beta1_yellow[8] 0.000 0.000 0.000 0.000 0.000
beta1_yellow[9] 0.000 0.000 0.000 0.000 0.000
beta1_yellow[10] 0.000 0.000 0.000 0.000 0.000
beta1_yellow[11] 2.092 0.427 1.294 2.087 2.947
beta1_yellow[12] 2.490 0.448 1.716 2.466 3.482
beta1_yellow[13] 2.809 0.459 1.997 2.782 3.785
beta1_yellow[14] 2.162 0.529 0.886 2.187 3.139
beta1_yellow[15] 2.107 0.399 1.384 2.089 2.939
beta1_yellow[16] 2.113 0.445 1.185 2.129 2.941
beta2_yellow[1] -3.630 3.126 -11.239 -2.876 -0.041
beta2_yellow[2] -3.625 3.118 -11.505 -2.766 -0.178
beta2_yellow[3] -3.435 3.036 -10.921 -2.640 -0.123
beta2_yellow[4] -3.015 3.098 -10.849 -2.013 -0.096
beta2_yellow[5] 0.000 0.000 0.000 0.000 0.000
beta2_yellow[6] 0.000 0.000 0.000 0.000 0.000
beta2_yellow[7] 0.000 0.000 0.000 0.000 0.000
beta2_yellow[8] 0.000 0.000 0.000 0.000 0.000
beta2_yellow[9] 0.000 0.000 0.000 0.000 0.000
beta2_yellow[10] 0.000 0.000 0.000 0.000 0.000
beta2_yellow[11] -4.402 2.494 -10.361 -3.912 -1.038
beta2_yellow[12] -4.844 2.700 -11.796 -4.219 -1.440
beta2_yellow[13] -4.391 2.258 -10.373 -3.899 -1.429
beta2_yellow[14] -4.640 2.810 -11.643 -4.119 -0.250
beta2_yellow[15] -4.230 2.622 -10.967 -3.569 -0.966
beta2_yellow[16] -4.877 2.701 -11.963 -4.260 -1.369
beta3_yellow[1] 25.774 7.095 18.252 22.644 43.956
beta3_yellow[2] 29.066 1.818 25.182 28.889 32.807
beta3_yellow[3] 32.888 3.123 25.093 32.834 39.830
beta3_yellow[4] 28.872 3.452 21.320 27.974 35.986
beta3_yellow[5] 29.841 7.815 18.473 28.660 44.804
beta3_yellow[6] 29.956 7.852 18.497 28.985 44.566
beta3_yellow[7] 30.014 7.799 18.451 29.299 44.897
beta3_yellow[8] 29.859 7.881 18.425 28.989 44.740
beta3_yellow[9] 30.038 7.982 18.419 29.264 44.781
beta3_yellow[10] 30.303 7.999 18.483 29.608 44.965
beta3_yellow[11] 45.284 0.543 43.990 45.376 45.974
beta3_yellow[12] 43.306 0.376 42.543 43.290 44.026
beta3_yellow[13] 44.835 0.391 43.981 44.907 45.501
beta3_yellow[14] 43.861 2.079 35.188 44.192 45.839
beta3_yellow[15] 45.158 0.521 44.139 45.147 45.958
beta3_yellow[16] 44.565 0.646 43.427 44.547 45.863
mu_beta0_yellow[1] 0.095 0.556 -1.075 0.088 1.200
mu_beta0_yellow[2] 0.637 0.336 -0.115 0.659 1.238
mu_beta0_yellow[3] -2.415 0.630 -3.393 -2.505 -0.945
tau_beta0_yellow[1] 1.692 1.869 0.093 1.154 6.666
tau_beta0_yellow[2] 3.464 3.847 0.310 2.420 13.260
tau_beta0_yellow[3] 1.296 1.596 0.104 0.852 5.005
beta0_black[1] -0.077 0.161 -0.394 -0.079 0.235
beta0_black[2] 1.912 0.128 1.661 1.913 2.170
beta0_black[3] 1.318 0.137 1.047 1.318 1.586
beta0_black[4] 2.428 0.131 2.172 2.425 2.689
beta0_black[5] 4.611 2.082 1.766 4.179 9.781
beta0_black[6] 4.650 1.933 2.270 4.179 9.823
beta0_black[7] 3.759 1.875 1.537 3.271 9.181
beta0_black[8] 0.959 0.212 0.555 0.958 1.378
beta0_black[9] 2.609 0.230 2.157 2.604 3.063
beta0_black[10] 1.457 0.133 1.201 1.457 1.716
beta0_black[11] 3.481 0.155 3.195 3.478 3.792
beta0_black[12] 4.865 0.174 4.529 4.865 5.208
beta0_black[13] -0.139 0.270 -0.688 -0.126 0.332
beta0_black[14] 2.849 0.161 2.529 2.847 3.166
beta0_black[15] 1.297 0.154 0.996 1.297 1.598
beta0_black[16] 4.274 0.164 3.956 4.271 4.606
beta2_black[1] 7.790 9.952 0.539 3.554 38.408
beta2_black[2] 0.000 0.000 0.000 0.000 0.000
beta2_black[3] 0.000 0.000 0.000 0.000 0.000
beta2_black[4] 0.000 0.000 0.000 0.000 0.000
beta2_black[5] 0.000 0.000 0.000 0.000 0.000
beta2_black[6] 0.000 0.000 0.000 0.000 0.000
beta2_black[7] 0.000 0.000 0.000 0.000 0.000
beta2_black[8] 0.000 0.000 0.000 0.000 0.000
beta2_black[9] 0.000 0.000 0.000 0.000 0.000
beta2_black[10] 0.000 0.000 0.000 0.000 0.000
beta2_black[11] 0.000 0.000 0.000 0.000 0.000
beta2_black[12] 0.000 0.000 0.000 0.000 0.000
beta2_black[13] -1.751 1.512 -6.081 -1.283 -0.272
beta2_black[14] 0.000 0.000 0.000 0.000 0.000
beta2_black[15] 0.000 0.000 0.000 0.000 0.000
beta2_black[16] 0.000 0.000 0.000 0.000 0.000
beta3_black[1] 41.802 1.097 39.778 41.959 43.367
beta3_black[2] 25.000 0.000 25.000 25.000 25.000
beta3_black[3] 25.000 0.000 25.000 25.000 25.000
beta3_black[4] 25.000 0.000 25.000 25.000 25.000
beta3_black[5] 25.000 0.000 25.000 25.000 25.000
beta3_black[6] 25.000 0.000 25.000 25.000 25.000
beta3_black[7] 25.000 0.000 25.000 25.000 25.000
beta3_black[8] 25.000 0.000 25.000 25.000 25.000
beta3_black[9] 25.000 0.000 25.000 25.000 25.000
beta3_black[10] 25.000 0.000 25.000 25.000 25.000
beta3_black[11] 25.000 0.000 25.000 25.000 25.000
beta3_black[12] 25.000 0.000 25.000 25.000 25.000
beta3_black[13] 39.072 1.410 36.724 39.267 40.654
beta3_black[14] 25.000 0.000 25.000 25.000 25.000
beta3_black[15] 25.000 0.000 25.000 25.000 25.000
beta3_black[16] 25.000 0.000 25.000 25.000 25.000
beta4_black[1] -0.259 0.194 -0.640 -0.261 0.120
beta4_black[2] 0.247 0.185 -0.122 0.248 0.614
beta4_black[3] -0.932 0.196 -1.325 -0.934 -0.542
beta4_black[4] 0.425 0.220 -0.005 0.423 0.855
beta4_black[5] 0.544 1.438 -1.223 0.309 3.702
beta4_black[6] 0.543 1.410 -1.286 0.317 3.698
beta4_black[7] 0.421 1.200 -1.410 0.257 3.199
beta4_black[8] -0.247 0.314 -0.883 -0.237 0.353
beta4_black[9] 0.821 0.768 -0.272 0.680 2.722
beta4_black[10] 0.051 0.185 -0.320 0.053 0.420
beta4_black[11] -0.689 0.214 -1.107 -0.684 -0.280
beta4_black[12] 0.169 0.325 -0.441 0.162 0.821
beta4_black[13] -1.190 0.222 -1.636 -1.183 -0.762
beta4_black[14] -0.176 0.246 -0.652 -0.180 0.309
beta4_black[15] -0.895 0.216 -1.322 -0.896 -0.470
beta4_black[16] -0.595 0.234 -1.046 -0.595 -0.133
mu_beta0_black[1] 1.289 0.884 -0.630 1.326 2.986
mu_beta0_black[2] 2.700 1.061 0.774 2.589 5.086
mu_beta0_black[3] 2.476 1.004 0.336 2.536 4.365
tau_beta0_black[1] 0.635 0.588 0.056 0.454 2.157
tau_beta0_black[2] 0.463 0.637 0.046 0.252 2.203
tau_beta0_black[3] 0.230 0.152 0.050 0.191 0.626
beta0_dsr[11] -2.888 0.295 -3.495 -2.888 -2.327
beta0_dsr[12] 4.554 0.278 3.995 4.557 5.090
beta0_dsr[13] -1.368 0.355 -2.015 -1.355 -0.774
beta0_dsr[14] -3.659 0.517 -4.626 -3.658 -2.620
beta0_dsr[15] -1.943 0.279 -2.490 -1.949 -1.401
beta0_dsr[16] -3.000 0.369 -3.727 -2.986 -2.303
beta1_dsr[11] 4.825 0.307 4.235 4.821 5.421
beta1_dsr[12] 10.707 55.183 2.282 5.049 22.960
beta1_dsr[13] 2.894 0.428 2.253 2.864 3.604
beta1_dsr[14] 6.328 0.543 5.230 6.339 7.346
beta1_dsr[15] 3.339 0.283 2.788 3.338 3.893
beta1_dsr[16] 5.814 0.383 5.080 5.817 6.564
beta2_dsr[11] -8.345 2.402 -14.088 -8.003 -4.710
beta2_dsr[12] -7.141 2.620 -12.849 -6.968 -2.503
beta2_dsr[13] -6.403 2.844 -12.529 -6.348 -0.701
beta2_dsr[14] -6.147 2.714 -11.938 -5.947 -1.875
beta2_dsr[15] -7.783 2.440 -13.347 -7.463 -3.925
beta2_dsr[16] -7.982 2.365 -13.359 -7.630 -4.264
beta3_dsr[11] 43.487 0.152 43.210 43.484 43.767
beta3_dsr[12] 33.973 0.728 32.154 34.115 34.810
beta3_dsr[13] 43.248 0.359 42.740 43.195 43.896
beta3_dsr[14] 43.347 0.240 43.071 43.278 43.964
beta3_dsr[15] 43.503 0.185 43.172 43.505 43.843
beta3_dsr[16] 43.441 0.163 43.164 43.426 43.760
beta4_dsr[11] 0.582 0.220 0.162 0.579 1.024
beta4_dsr[12] 0.231 0.441 -0.659 0.226 1.088
beta4_dsr[13] -0.169 0.222 -0.598 -0.166 0.256
beta4_dsr[14] 0.146 0.252 -0.359 0.145 0.644
beta4_dsr[15] 0.726 0.217 0.325 0.716 1.166
beta4_dsr[16] 0.152 0.226 -0.280 0.153 0.593
beta0_slope[11] -1.847 0.150 -2.139 -1.846 -1.560
beta0_slope[12] -4.464 0.254 -4.970 -4.460 -3.966
beta0_slope[13] -1.342 0.182 -1.746 -1.328 -1.037
beta0_slope[14] -2.678 0.165 -3.008 -2.674 -2.368
beta0_slope[15] -1.340 0.146 -1.629 -1.341 -1.056
beta0_slope[16] -2.735 0.154 -3.027 -2.740 -2.427
beta1_slope[11] 4.486 0.225 4.051 4.490 4.916
beta1_slope[12] 3.980 0.448 3.108 3.978 4.843
beta1_slope[13] 2.714 0.440 2.206 2.647 3.972
beta1_slope[14] 6.323 0.413 5.519 6.319 7.124
beta1_slope[15] 3.009 0.210 2.593 3.010 3.414
beta1_slope[16] 5.286 0.284 4.738 5.283 5.849
beta2_slope[11] 8.551 2.267 5.055 8.274 13.768
beta2_slope[12] 6.600 2.858 1.218 6.647 12.303
beta2_slope[13] 5.264 3.015 0.395 5.120 11.339
beta2_slope[14] 6.325 2.554 2.199 6.115 11.829
beta2_slope[15] 8.122 2.328 4.560 7.843 13.561
beta2_slope[16] 7.648 2.275 4.312 7.287 13.271
beta3_slope[11] 43.459 0.134 43.225 43.455 43.734
beta3_slope[12] 43.355 0.289 42.900 43.305 43.925
beta3_slope[13] 43.457 0.393 42.879 43.407 44.059
beta3_slope[14] 43.271 0.139 43.094 43.236 43.623
beta3_slope[15] 43.490 0.164 43.195 43.488 43.802
beta3_slope[16] 43.370 0.140 43.154 43.350 43.687
beta4_slope[11] -0.730 0.164 -1.069 -0.727 -0.413
beta4_slope[12] -1.151 0.466 -2.153 -1.107 -0.352
beta4_slope[13] 0.091 0.163 -0.231 0.095 0.415
beta4_slope[14] -0.095 0.197 -0.467 -0.096 0.283
beta4_slope[15] -0.759 0.159 -1.078 -0.756 -0.450
beta4_slope[16] -0.168 0.175 -0.531 -0.167 0.175
sigma_H[1] 0.199 0.052 0.101 0.198 0.311
sigma_H[2] 0.172 0.030 0.120 0.170 0.239
sigma_H[3] 0.196 0.042 0.122 0.192 0.285
sigma_H[4] 0.419 0.075 0.296 0.411 0.584
sigma_H[5] 0.997 0.209 0.620 0.987 1.437
sigma_H[6] 0.392 0.202 0.031 0.384 0.817
sigma_H[7] 0.307 0.062 0.212 0.300 0.451
sigma_H[8] 0.416 0.094 0.279 0.405 0.610
sigma_H[9] 0.529 0.127 0.330 0.513 0.821
sigma_H[10] 0.215 0.041 0.144 0.212 0.304
sigma_H[11] 0.277 0.045 0.199 0.274 0.376
sigma_H[12] 0.439 0.166 0.208 0.416 0.782
sigma_H[13] 0.216 0.039 0.149 0.212 0.301
sigma_H[14] 0.508 0.091 0.346 0.502 0.698
sigma_H[15] 0.246 0.040 0.180 0.242 0.335
sigma_H[16] 0.223 0.042 0.155 0.219 0.317
lambda_H[1] 3.179 4.326 0.146 1.829 13.481
lambda_H[2] 8.332 8.276 0.836 6.063 28.086
lambda_H[3] 6.128 9.167 0.263 3.019 30.426
lambda_H[4] 0.006 0.004 0.001 0.005 0.018
lambda_H[5] 4.365 11.477 0.038 1.128 30.304
lambda_H[6] 7.916 16.405 0.008 0.817 59.636
lambda_H[7] 0.013 0.009 0.002 0.011 0.036
lambda_H[8] 8.299 10.639 0.135 4.620 36.539
lambda_H[9] 0.015 0.010 0.003 0.013 0.040
lambda_H[10] 0.321 0.747 0.036 0.206 1.192
lambda_H[11] 0.262 0.395 0.011 0.128 1.358
lambda_H[12] 4.590 5.951 0.181 2.671 20.462
lambda_H[13] 3.509 3.165 0.240 2.658 11.681
lambda_H[14] 3.289 3.805 0.224 2.077 12.447
lambda_H[15] 0.025 0.044 0.003 0.016 0.095
lambda_H[16] 0.846 1.354 0.044 0.441 4.024
mu_lambda_H[1] 4.319 1.946 1.171 4.159 8.480
mu_lambda_H[2] 3.826 1.957 0.604 3.667 8.082
mu_lambda_H[3] 3.439 1.819 0.780 3.122 7.729
sigma_lambda_H[1] 8.577 4.380 2.035 7.973 18.290
sigma_lambda_H[2] 8.327 4.723 0.974 7.789 18.396
sigma_lambda_H[3] 6.150 3.932 1.060 5.297 16.662
beta_H[1,1] 6.914 1.083 4.333 7.073 8.529
beta_H[2,1] 9.888 0.483 8.809 9.906 10.817
beta_H[3,1] 7.979 0.776 6.046 8.094 9.204
beta_H[4,1] 9.548 7.934 -6.325 9.736 25.235
beta_H[5,1] 0.149 2.305 -4.626 0.358 4.078
beta_H[6,1] 3.135 3.990 -6.905 4.509 7.627
beta_H[7,1] 0.586 5.697 -11.567 0.866 10.736
beta_H[8,1] 1.292 3.450 -2.484 1.229 3.563
beta_H[9,1] 13.139 5.730 1.961 13.029 24.654
beta_H[10,1] 7.058 1.653 3.636 7.124 10.256
beta_H[11,1] 5.071 3.500 -2.871 5.913 9.914
beta_H[12,1] 2.624 1.069 0.686 2.566 4.998
beta_H[13,1] 9.054 0.850 7.255 9.118 10.391
beta_H[14,1] 2.168 1.001 0.179 2.163 4.087
beta_H[15,1] -6.191 3.733 -12.856 -6.417 1.815
beta_H[16,1] 3.467 2.708 -0.862 3.169 9.833
beta_H[1,2] 7.909 0.239 7.423 7.913 8.354
beta_H[2,2] 10.027 0.134 9.770 10.025 10.284
beta_H[3,2] 8.950 0.198 8.567 8.949 9.353
beta_H[4,2] 3.545 1.472 0.747 3.503 6.471
beta_H[5,2] 1.953 0.972 -0.036 1.976 3.778
beta_H[6,2] 5.719 1.080 3.201 5.908 7.349
beta_H[7,2] 2.660 1.095 0.690 2.590 4.994
beta_H[8,2] 3.008 1.046 1.392 3.124 4.215
beta_H[9,2] 3.463 1.103 1.349 3.433 5.676
beta_H[10,2] 8.207 0.342 7.495 8.216 8.861
beta_H[11,2] 9.773 0.632 8.849 9.645 11.197
beta_H[12,2] 3.936 0.380 3.178 3.929 4.707
beta_H[13,2] 9.123 0.247 8.654 9.115 9.635
beta_H[14,2] 4.021 0.357 3.328 4.008 4.742
beta_H[15,2] 11.376 0.671 9.938 11.405 12.623
beta_H[16,2] 4.520 0.800 3.015 4.508 6.146
beta_H[1,3] 8.459 0.234 8.039 8.445 8.957
beta_H[2,3] 10.063 0.116 9.834 10.063 10.295
beta_H[3,3] 9.615 0.163 9.296 9.613 9.954
beta_H[4,3] -2.508 0.878 -4.227 -2.493 -0.835
beta_H[5,3] 3.824 0.614 2.541 3.847 4.990
beta_H[6,3] 8.022 1.190 6.379 7.644 10.613
beta_H[7,3] -2.802 0.654 -4.089 -2.796 -1.579
beta_H[8,3] 5.250 0.480 4.642 5.194 6.219
beta_H[9,3] -2.836 0.730 -4.282 -2.830 -1.448
beta_H[10,3] 8.685 0.271 8.161 8.681 9.219
beta_H[11,3] 8.536 0.287 7.927 8.556 9.047
beta_H[12,3] 5.250 0.322 4.464 5.293 5.775
beta_H[13,3] 8.846 0.175 8.483 8.851 9.178
beta_H[14,3] 5.713 0.277 5.103 5.733 6.192
beta_H[15,3] 10.370 0.314 9.767 10.365 11.007
beta_H[16,3] 6.241 0.597 4.909 6.302 7.255
beta_H[1,4] 8.263 0.181 7.874 8.272 8.598
beta_H[2,4] 10.130 0.119 9.879 10.136 10.348
beta_H[3,4] 10.114 0.167 9.747 10.126 10.411
beta_H[4,4] 11.795 0.444 10.909 11.789 12.686
beta_H[5,4] 5.461 0.736 4.274 5.368 7.177
beta_H[6,4] 7.031 0.943 4.944 7.322 8.313
beta_H[7,4] 8.289 0.352 7.597 8.290 8.994
beta_H[8,4] 6.717 0.248 6.246 6.726 7.143
beta_H[9,4] 7.204 0.472 6.273 7.197 8.126
beta_H[10,4] 7.760 0.233 7.325 7.747 8.240
beta_H[11,4] 9.389 0.201 8.998 9.391 9.779
beta_H[12,4] 7.144 0.214 6.758 7.136 7.596
beta_H[13,4] 9.041 0.145 8.746 9.048 9.311
beta_H[14,4] 7.727 0.222 7.289 7.721 8.178
beta_H[15,4] 9.468 0.237 9.020 9.467 9.940
beta_H[16,4] 9.338 0.237 8.902 9.325 9.830
beta_H[1,5] 8.994 0.145 8.696 8.998 9.273
beta_H[2,5] 10.786 0.093 10.610 10.784 10.968
beta_H[3,5] 10.922 0.174 10.618 10.909 11.293
beta_H[4,5] 8.397 0.458 7.544 8.378 9.367
beta_H[5,5] 5.424 0.563 4.107 5.461 6.412
beta_H[6,5] 8.798 0.628 7.898 8.646 10.301
beta_H[7,5] 6.739 0.342 6.076 6.736 7.424
beta_H[8,5] 8.214 0.209 7.849 8.206 8.640
beta_H[9,5] 8.199 0.476 7.270 8.199 9.155
beta_H[10,5] 10.086 0.227 9.609 10.086 10.517
beta_H[11,5] 11.506 0.225 11.063 11.508 11.944
beta_H[12,5] 8.482 0.200 8.085 8.480 8.895
beta_H[13,5] 10.007 0.130 9.758 10.006 10.272
beta_H[14,5] 9.199 0.238 8.772 9.186 9.693
beta_H[15,5] 11.174 0.245 10.680 11.175 11.642
beta_H[16,5] 9.912 0.180 9.548 9.914 10.255
beta_H[1,6] 10.176 0.187 9.841 10.163 10.575
beta_H[2,6] 11.513 0.108 11.303 11.511 11.730
beta_H[3,6] 10.814 0.163 10.454 10.826 11.104
beta_H[4,6] 12.876 0.806 11.196 12.892 14.378
beta_H[5,6] 5.879 0.592 4.726 5.862 7.040
beta_H[6,6] 8.764 0.688 6.960 8.888 9.738
beta_H[7,6] 9.889 0.572 8.819 9.884 11.044
beta_H[8,6] 9.519 0.273 9.041 9.529 9.947
beta_H[9,6] 8.466 0.789 6.993 8.450 10.090
beta_H[10,6] 9.514 0.313 8.845 9.545 10.064
beta_H[11,6] 10.810 0.352 10.040 10.839 11.437
beta_H[12,6] 9.376 0.260 8.867 9.366 9.909
beta_H[13,6] 11.048 0.169 10.746 11.041 11.411
beta_H[14,6] 9.820 0.291 9.237 9.829 10.364
beta_H[15,6] 10.829 0.431 9.988 10.830 11.690
beta_H[16,6] 10.522 0.240 10.012 10.530 10.988
beta_H[1,7] 10.864 0.862 8.679 10.965 12.249
beta_H[2,7] 12.215 0.430 11.306 12.225 13.039
beta_H[3,7] 10.535 0.660 9.071 10.579 11.657
beta_H[4,7] 2.520 4.189 -5.213 2.381 11.037
beta_H[5,7] 6.389 1.752 3.013 6.359 10.299
beta_H[6,7] 9.565 2.507 4.498 9.518 16.106
beta_H[7,7] 10.411 2.871 4.764 10.444 16.006
beta_H[8,7] 10.942 0.983 9.380 10.902 12.560
beta_H[9,7] 4.515 4.005 -3.644 4.634 12.152
beta_H[10,7] 9.813 1.449 7.155 9.703 12.949
beta_H[11,7] 11.032 1.715 7.866 10.888 14.885
beta_H[12,7] 9.983 0.987 7.756 10.064 11.564
beta_H[13,7] 11.652 0.767 9.921 11.743 12.872
beta_H[14,7] 10.381 0.946 8.342 10.444 12.033
beta_H[15,7] 12.040 2.252 7.556 12.047 16.338
beta_H[16,7] 12.302 1.255 10.174 12.142 15.106
beta0_H[1] 8.626 13.139 -18.178 8.631 35.204
beta0_H[2] 10.612 6.386 -3.235 10.709 24.160
beta0_H[3] 9.719 10.295 -9.875 9.799 28.883
beta0_H[4] 10.321 180.823 -346.937 7.579 403.966
beta0_H[5] 4.317 24.554 -40.809 4.254 51.540
beta0_H[6] 5.769 55.249 -117.433 7.387 119.390
beta0_H[7] 5.252 134.320 -271.868 2.121 264.322
beta0_H[8] 6.333 25.454 -16.532 6.397 25.615
beta0_H[9] 5.156 124.068 -242.427 5.481 243.516
beta0_H[10] 8.525 31.676 -57.019 8.262 73.555
beta0_H[11] 10.674 50.092 -95.814 11.119 120.251
beta0_H[12] 6.902 11.532 -16.680 6.842 29.365
beta0_H[13] 10.035 10.925 -10.475 9.893 30.304
beta0_H[14] 7.018 11.553 -15.812 7.024 31.098
beta0_H[15] 8.636 106.948 -207.834 8.119 218.083
beta0_H[16] 8.501 25.204 -44.559 8.802 59.810
## [1] 10